Homework Solutions

Geometry CP, Jan 24

Rectangles, Rhombi, and Squares

Book Section: 6-4 & 6-5

Essential Question: What special case parallelograms should I know,

and what are their characteristics?

Standards: CCSS G.CO.10, G.MG.3; G-3.5

Rectangles • Rectangle – A parallelogram with four right angles.

• A rectangle inherits all parallelogram properties, and it

has four right angles.

Opposite sides are parallel and congruent

Opposite angles are congruent

Consecutive angles are supplementary

Diagonals bisect each other

It has four right angles

And if that wasn’t enough, there’s more!

Rectangle Diagonal Theorem

Example 1 Use Properties of Rectangles

CONSTRUCTION A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet, and LN = 6.5 feet, find KM.

Example 2

A. 3 feet

B. 7.5 feet

C. 9 feet

D. 12 feet

Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.

The Other Way on Rectangles

Example 3 Proving Rectangle Relationships

ART Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides and the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, DA = 35 inches, BD = 37 inches, and AC = 37 inches, explain how an artist can be sure that the frame is rectangular.

Example 4

Max is building a swimming pool in his backyard. He measures the length and width of the pool so that opposite sides are parallel. He also measures the diagonals of the pool to make sure that they are congruent. How does he know that the measure of each corner is 90?

A. Since opp. sides are ||, STUR must be a rectangle.

B. Since opp. sides are , STUR must be a rectangle.

C. Since diagonals of the are , STUR must be a rectangle.

D. STUR is not a rectangle.

Example 5 Rectangles and Coordinate Geometry

Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3). Determine whether JKLM is a rectangle using the Distance Formula.

Example 6

Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1). What are the lengths of diagonals WY and XZ?

A.

B. 4

C. 5

D. 25

Rhombus • Rhombus – A parallelogram with four congruent sides.

• A rhombus inherits all parallelogram properties, and it

has four congruent sides.

Opposite sides are parallel and congruent

Opposite angles are congruent

Consecutive angles are supplementary

Diagonals bisect each other

It has four congruent sides

And if that wasn’t enough, there’s more here too!

Rhombus Diagonal Properties

Example 7 Use Properties of a Rhombus

A. The diagonals of rhombus WXYZ intersect at V. If mWZX = 39.5, find mZYX.

Example 8 Use Properties of a Rhombus

B. ALGEBRA The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.

Example 9

A. mCDB = 126

B. mCDB = 63

C. mCDB = 54

D. mCDB = 27

A. ABCD is a rhombus. Find mCDB if mABC = 126.

Example 10

A. x = 1

B. x = 3

C. x = 4

D. x = 6

B. ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x.

Square • Square – A rhombus with four right angles.

• A square inherits all properties of its successors, and it

has four congruent sides and four right angles.

Opposite sides are parallel and congruent

Opposite angles are congruent

Consecutive angles are supplementary

Diagonals bisect each other, and intersect at right angles

It has four congruent sides

It has four right angles

And it keeps getting better and better!

Rhombi and Squares

The Parallelogram Family Portrait

Example 11

Is there enough information given to prove that ABCD is a rhombus?

Given: ABCD is a parallelogram. AD DC

Prove: ADCD is a rhombus

Example 12 Use Conditions for Rhombi and

Squares

GARDENING Hector is measuring the boundary of a new garden. He wants the garden to be square. He has set each of the corner stakes 6 feet apart. What does Hector need to know to make sure that the garden is square?

Example 13

A. The diagonal bisects a pair of opposite angles.

B. The diagonals bisect each other.

C. The diagonals are perpendicular.

D. The diagonals are congruent.

Sachin has a shape he knows to be a parallelogram and all four sides are congruent. Which information does he need to know to determine whether it is also a square?

Examples

Classwork: Textbook p.426-427, 10-13 &

26-31; p.435, 7-12, 9-14, 24-25

Homework: HW Due 1/28, Parts 1 and 2